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Let X₁, X2, and X3 represent the times necessary to perform three successive repairs tasks at a certain service facility. Suppose they are independent, normal random variables with expected and variances o,o2, and o3, respectively values M1, M₂₂ a) If μ = μ₂=₂= 60 and o² = 0² = 03 =15, calculate P(X₁ + X₂ + X3 ≤ 200). b) Using the u's and o's given in part (a), calculate P(58 < X < 62), where and | Из

Let X₁, X2, and X3 represent the times necessary to perform three successive repairs tasks at a certain service facility. Suppose they are independent, normal random variables with expected and variances o,o2, and o3, respectively values M1, M₂₂ a) If μ = μ₂=₂= 60 and o² = 0² = 03 =15, calculate P(X₁ + X₂ + X3 ≤ 200). b) Using the u's and o's given in part (a), calculate P(58 < X < 62), where and | Из

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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1. Let \( X_1, X_2, \) and \( X_3 \) represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal random variables with expected values \( \mu_1, \mu_2, \) and \( \mu_3 \) and variances \( \sigma_1^2, \sigma_2^2, \) and \( \sigma_3^2 \) respectively. a) If \( \mu_1 = \mu_2 = \mu_3 = 60 \) and \( \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 15 \), calculate \( P(X_1 + X_2 + X_3 \leq 200) \). b) Using the \(\mu\)’s and \(\sigma\)’s given in part (a), calculate \( P(58 \leq \overline{X} \leq 62) \), where \[ \overline{X} = \frac{X_1 + X_2 + X_3}{3}. \] c) Using the \(\mu\)’s and \(\sigma\)’s given in part (a), calculate \( P(-10 \leq X_1 - 0.5X_2 - 0.5X_3 \leq 5) \). d) If \( \mu_1 = 40, \ \mu_2 = 50, \ \mu_3 = 60, \ \sigma_1^2 = 10, \ \sigma_2^2 = 12, \) and \( \sigma_3^2 = 14 \), calculate \( P(X_1 + X_2 - 2X_3 \geq 0) \).
Transcribed Image Text:1. Let \( X_1, X_2, \) and \( X_3 \) represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal random variables with expected values \( \mu_1, \mu_2, \) and \( \mu_3 \) and variances \( \sigma_1^2, \sigma_2^2, \) and \( \sigma_3^2 \) respectively. a) If \( \mu_1 = \mu_2 = \mu_3 = 60 \) and \( \sigma_1^2 = \sigma_2^2 = \sigma_3^2 = 15 \), calculate \( P(X_1 + X_2 + X_3 \leq 200) \). b) Using the \(\mu\)’s and \(\sigma\)’s given in part (a), calculate \( P(58 \leq \overline{X} \leq 62) \), where \[ \overline{X} = \frac{X_1 + X_2 + X_3}{3}. \] c) Using the \(\mu\)’s and \(\sigma\)’s given in part (a), calculate \( P(-10 \leq X_1 - 0.5X_2 - 0.5X_3 \leq 5) \). d) If \( \mu_1 = 40, \ \mu_2 = 50, \ \mu_3 = 60, \ \sigma_1^2 = 10, \ \sigma_2^2 = 12, \) and \( \sigma_3^2 = 14 \), calculate \( P(X_1 + X_2 - 2X_3 \geq 0) \).
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